Approaching quantum limits in sub-Rayleigh resolution using phase modulation optimization
Spatial-mode demultiplexing (SPADE) has made significant progress in incoherent optical imaging and has been implemented through various experimental approaches. Holography-based SPADE is widely used in proof-of-principle experiments because of its low cost, stable optical arrangement, and ease of implementation. However, its low photon utilization severely limits its practical applications. Alternating Fourier transforms and phase modulations can implement unitary optical transformations. Building on this idea, we use backpropagation to optimize the phase profiles of spatial light modulators (SLMs) and thereby design quantum measurements that are efficient in both information and energy. We study centroid--separation estimation for two incoherent point sources, using Fisher information or the estimation errors of an auxiliary neural network estimator as optimization objectives. As the number of SLMs increases, the optimized measurements approach the quantum limits characterized by the quantum Cramér-Rao bound and the information-regret tradeoff relation. With a few SLMs, our method can outperform direct imaging, showing its potential for superresolution imaging.